Tony Shaska

Integral minimal models for binary forms


June 5, 2015, 1:55 — 2:45pm at LIT 368.


Let \(K\) be a number field, \(\mathfrak O_K\) its ring of integers, and \(f \in \mathfrak O_K [x, z]\) a degree \(n\) binary form.

We devise an algorithm which accomplishes the following:


  1. finds a binary form \(f^\prime\), \(\bar K\)-equivalent to \(f\), with minimal height \(h (f)\) (i.e., smallest coefficients),
  2. enumerates all twists \(g \in O_K [x, z]\) of \(f\) with \(h (g) \leq h(f)\).

Our methods rely and  extend previous results of Hermite, Julia, Cremona, and Stoll.